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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"57627d4cd44f104a6d921e6db6590c50399922f6a6e456cc8e741f16929b8795","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-5d7c3f4772a9d8bb","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      1;\n  1:      1;\n  2:      2,       4;\n  3:      6,      20;\n  4:     24,     112,      45;\n  5:    120,     720,     549;\n  6:    720,    5280,    5958,     976;\n  7:   5040,   43680,   65394,   22160;\n  8:  40320,  403200,  752976,  393088,   34625;\n  9: 362880, 4112640, 9208080, 6533760, 1263025;\n","recurrence_sentence":"The row polynomials satisfy n^2*P(n,x) = -(n^4+n^3-n^2-n+(4*n^4-8*n^3+4*n^2-16)*x-(9*n^4-13*n^3-10*n^2)*x^2)*P(n-2,x) + ((4*n^3-2*n^2+6)*x+(4*n^4-20*n^2+20)*x^2+(4*n^4-32*n^3-18*n^2)*x^3)*d/dx P(n-2,x) - ((4*n^2-4)*x^2+(8*n^3+8*n^2-4)*x^3+(8*n^3+4*n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^3+2*n^2+n+(4*n^3-4*n^2)*x)*P(n-1,x) - ((4*n^2-2)*x+(4*n^3+4*n^2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^4 + n^3 - n^2 - n + (4*n^4 - 8*n^3 + 4*n^2 - 16)*x - (9*n^4 - 13*n^3 - 10*n^2)*x^2)*P[n - 2, x] + ((4*n^3 - 2*n^2 + 6)*x + (4*n^4 - 20*n^2 + 20)*x^2 + (4*n^4 - 32*n^3 - 18*n^2)*x^3)*D[P[n - 2, x], x] - ((4*n^2 - 4)*x^2 + (8*n^3 + 8*n^2 - 4)*x^3 + (8*n^3 + 4*n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^3 + 2*n^2 + n + (4*n^3 - 4*n^2)*x)*P[n - 1, x] - ((4*n^2 - 2)*x + (4*n^3 + 4*n^2)*x^2)*D[P[n - 1, x], x])/(n^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 + n^3 - n^2 - n + (4*n^4 - 8*n^3 + 4*n^2 - 16)*x - (9*n^4 - 13*n^3 - 10*n^2)*x^2)*v[i - 2] + ((4*n^3 - 2*n^2 + 6)*x + (4*n^4 - 20*n^2 + 20)*x^2 + (4*n^4 - 32*n^3 - 18*n^2)*x^3)*deriv(v[i - 2], x) - ((4*n^2 - 4)*x^2 + (8*n^3 + 8*n^2 - 4)*x^3 + (8*n^3 + 4*n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^3 + 2*n^2 + n + (4*n^3 - 4*n^2)*x)*v[i - 1] - ((4*n^2 - 2)*x + (4*n^3 + 4*n^2)*x^2)*deriv(v[i - 1], x))/(n^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-d8312bb744732845","steps":["\u03b3 (scale 1/4)","x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-16125ab4d60fcfd7","e-42ac1b21fb8d0456","e-2cf78bdb41a0e81c"],"text":"type_b_eulerian \u2192 \u03b3 (scale 1/4) \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"type_b_eulerian \u2192 \u03b3 (scale 1/4) \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}